Long-range dependencies in heart rate signals- revisited

The RR series extracted from human electrocardiogram signal (ECG) is considered as a fractal stochastic process. The manifestation of long-range dependencies is the presence of power laws in scale dependent process characteristics. Exponents of these laws: $\beta$ - describing power spectrum decay, $\alpha$ - responsible for decay of detrended fluctuations or $H$ related to, so-called, roughness of a signal, are known to differentiate hearts of healthy people from hearts with congestive heart failure. There is a strong expectation that resolution spectrum of exponents, so-called, local exponents in place of global exponents allows to study differences between hearts in details. The arguments are given that local exponents obtained in multifractal analysis by the two methods: wavelet transform modulus maxima (WTMM) and multifractal detrended fluctuation analysis (MDFA), allow to recognize the following four stages of the heart: healthy and young, healthy and advance in years, subjects with left ventricle systolic dysfunction (NYHA I--III class) and characterized by severe congestive heart failure (NYHA III-IV class).Comment: 24 page

Wavelet and Multiscale Analysis of Network Traffic

The complexity and richness of telecommunications traffic is such that one may despair to find any regularity or explanatory principles. Nonetheless, the discovery of scaling behaviour in tele-traffic has provided hope that parsimonious models can be found. The statistics of scaling behavior present many challenges, especially in non-stationary environments. In this paper we describe the state of the art in this area, focusing on the capabilities of the wavelet transform as a key tool for unravelling the mysteries of traffic statistics and dynamics

Self-organized model of cascade spreading

We study simultaneous price drops of real stocks and show that for high drop thresholds they follow a power-law distribution. To reproduce these collective downturns, we propose a minimal self-organized model of cascade spreading based on a probabilistic response of the system elements to stress conditions. This model is solvable using the theory of branching processes and the mean-field approximation. For a wide range of parameters, the system is in a critical state and displays a power-law cascade-size distribution similar to the empirically observed one. We further generalize the model to reproduce volatility clustering and other observed properties of real stocks.Comment: 8 pages, 6 figure

Multifractal products of stochastic processes: Construction and some basic properties

Abstract In various fields, such as teletraffic and economics, measured time series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a non-stationary process. To overcome this problem we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study L 2 -convergence, non-degeneracy and continuity of the limit process. Establishing a power law for its moments we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference

Random Subset Feature Selection for Ecological Niche Modeling of Wildfire Activity and the Monarch Butterfly

Correlative ecological niche models (ENMs) are essential for investigating distributions of species and natural phenomena via environmental correlates across broad fields, including entomology and pyrogeography featured in this study. Feature (variable) selection is critical for producing more robust ENMs with greater transferability across space and time, but few studies evaluate formal feature selection algorithms (FSAs) for producing higher performance ENMs. Variability of ENMs arising from feature subsets is also seldom represented. A novel FSA is developed and evaluated, the random subset feature selection algorithm (RSFSA). The RSFSA generates an ensemble of higher accuracy ENMs from different feature subsets, producing a feature subset ensemble (FSE). The RSFSA-selected FSEs are novelly used to represent ENM variability. Wildfire activity presence/absence databases for the western US prove ideal for evaluating RSFSA-selected MaxEnt ENMs. The RSFSA was effective in identifying FSEs of 15 of 90 variables with higher accuracy and information content than random FSEs. Selected FSEs were used to identify severe contemporary wildfire deficits and significant future increases in wildfire activity for many ecoregions. Migratory roosting localities of declining eastern North American monarch butterflies (Danaus plexippus) were used to spatially model migratory pathways, comparing RSFSAselected MaxEnt ENMs and kernel density estimate models (KDEMs). The higher information content ENMs best correlated migratory pathways with nectar resources in grasslands. Higher accuracy KDEMs best revealed migratory pathways through less suitable desert environments. Monarch butterfly roadkill data was surveyed for Texas within the main Oklahoma to Mexico Central Funnel migratory pathway. A random FSE of MaxEnt roadkill ENMs was used to estimate a 2-3% loss of migrants to roadkill. Hotspots of roadkill in west Texas and Mexico were recommended for assessing roadkill mitigation to assist in monarch population recovery. The RSFSA effectively produces higher performance ENM FSEs for estimating optimal feature subset sizes, and comparing ENM algorithms and parameters, and environmental scenarios. The RSFSA also performed comparably to expert variable selection, confirming its value in the absence of expert information. The RSFSA should be compared with other FSAs for developing ENMs and in data mining applications across other disciplines, such as image classification and molecular bioinformatics

Robustness of power-law behavior in cascading line failure models

Inspired by reliability issues in electric transmission networks, we use a probabilistic approach to study the occurrence of large failures in a stylized cascading line failure model. Such models capture the phenomenon where an initial line failure potentially triggers massive knock-on effects. Under certain critical conditions, the probability that the number of line failures exceeds a large threshold obeys a power-law distribution, a distinctive property observed in empiric blackout data. In this paper, we examine the robustness of the power-law behavior by exploring under which conditions this behavior prevails